Words and meanings
In maths, words can be very important and
sometimes have a more specific meaning than
we first think.
For example, if we talk about ‘doing a sum’,
many people think this means the same as a
‘doing a calculation’, but ‘sum’ means
‘addition’, so 7-3=4 is not strictly a ‘sum’ at all.
You will meet other words that are sometimes
misused in maths and science during these
activities.
Scalar and vector quantities
When talking about certain measures, there
are two types: scalars and vectors.
• A scalar quantity has magnitude (size)
• A vector quantity has magnitude and
direction
An example: If I start from home and walk at
5km per hour, how far from home am I after an
hour?
Scalar and vector quantities
Surely I must be 5km from home…
…but what if I turned around after half an
hour?
…what if the road isn’t a perfectly straight
one?
Scalar and vector quantities
If we assume that the road is perfectly
straight:
• distance travelled (a scalar quantity)
is how far I’ve walked, regardless of
whether I’ve turned around or not
• displacement (a vector quantity) is
how far I am along the road from my
starting point – in this case, home
Scalar and vector quantities
Vector quantities also have
direction, so if I walked in the
opposite direction along the road
from my house, it would be a
negative displacement.
Would the distance travelled also
be negative?
Scalar and vector quantities
We also sometimes use just ‘distance’
or ‘distance from xxxxxx’.
This is a scalar quantity and shows how
far from a certain object something is,
but takes no notice of direction.
Can you see that I can be
in two different positions
and still be the same
distance from the house?
Displacement
A displacement
- time graph
could look like
this:
What would the
distance-time
graph look like?
Displacement & Distance
Does one of the
graphs give you
a bit more
information
than the other?
Displacement & Distance
What would the
corresponding
distance
travelled-time
graph look like?
Displacement & Distance
Which of these is
most informative?
Displacement and distance
Which of the displacement-time graphs
on the next slide match with this
distance travelled-time one?
1
4
2
5
3
6
Displacement and distance
Which other pair of the displacementtime graphs would have the same
distance travelled-time graphs as each
other?
What would the distance travelled-time
graph for the third one look like?
3
1
4
Position
Another term used is ‘position’, this is
always relative to a specific object – in
this case the house.
Think about the two scenarios on the
next slide. For each one sketch:
• a position-time graph
• a displacement-time graph
• and ‘distance from the house’-time
graph
Position
1. I start from the house and walk in the
direction of the arrow at a steady
pace for 5 seconds, stop for 2
seconds, turn round and walk back
at a quicker pace for 5 seconds
2. I start from the red dot and walk in
the direction of the arrow at the
same steady pace, stop for 2
seconds, turn round and walk back
at the quicker pace for 5 seconds
Position
What do you notice about the graphs?
You should have noticed that the
displacement-time graphs are identical.
The position-time graphs have the same
shape as each other, and the same
shape as the displacement-time graphs,
but translated a little in each case.
Position
The ‘distance from the house’-time
graphs are similar to the others initially,
but when the walker gets to the other
side of the house, the graph is different
as it stays in the positive section of the
page (because it’s a scalar measure).
Speed and velocity
Speed and velocity are another pair of related
quantities:
• speed is a scalar quantity
• velocity is a vector quantity
Speed and velocity
A car travelling at a constant speed of 30km/h can
be going in any direction, whereas a car travelling
with a constant velocity of 30km/h means that the
car is moving in a specific direction.
Can speed, velocity or both
be negative?
Speed and velocity
Speed and velocity are both ‘rates of change’
which means they refer to how quickly
something is changing.
• Speed is the rate of change of distance
• Velocity is the rate of change of displacement
Speed and velocity
From these definitions we can infer that:
• the gradient of a distance travelled-time graph
is speed,
• the gradient of a displacement-time graph is
velocity.
Speed and velocity
Why would there be an issue with using the
gradient to work out speed from the ‘distance
from home’- time graph below?
Average speed and velocity
Average
speed =
Total distance travelled
Time
Average
velocity =
Displacement
(from start to finish)
Time
Are they the same?
Always?
Sketch a few graphs to convince
a partner.
Average speed and velocity
• Look at this
displacement-time graph.
• What is the displacement
from start to finish?
• What is the total distance
travelled?
• Work out the average
speed and the average
velocity.
Velocity, displacement and
distance travelled
A set of cards – copied on the next slide has 4 displacement-time graphs together
with the corresponding:
• Velocity-time graphs,
• Distance travelled
• Average speed
Match the sets and fill in the blanks
Average velocity is -1m/s
Average velocity is _______
Average velocity is 1.33m/s
Average velocity is 0m/s
Distance travelled is ________
Distance travelled is 12m
Distance travelled is 14m
Distance travelled is 14m
Average speed is _______
Average speed is 2m/s
Average speed is _______
Average speed is_________
Teacher notes: Kinematics
This edition looks at an introduction to 1 dimensional kinematics,
making it suitable for those undertaking the new GCSE and for those
starting Mechanics at A level.
The concept of scalar and vector quantities is introduced, and whilst it
is not a specific requirement within GCSE Mathematics to know the
terms and understand the distinction, it is in GCSE Science.
Additionally, both speed & velocity and displacement & distance are
referred to within the DfE objectives for KS4 mathematics and within
several exam board specifications.
Making references to both without explaining the difference to students
could cause confusion, particularly if they are encountering them in
Science, so parts of this activity address this issue.
Teacher notes: symbols
An opportunity for students to discuss
something in pairs and then feed back to the
class.
Students to write something down or work
something out.
A suggestion in the teacher notes of a way to
make an activity more ‘girl friendly’ in order to
increase the confidence of girls in class.
Teacher notes: Scalar and vector quantities
Slides 3 & 4: If I’ve walked 5km, I could
be anything from 0km to 5km from home.
We usually make assumptions in maths to
help simplify things, but these assumptions
are often not shared with students. With
‘distance-time’ work, our underlying
assumption is often that the road or path or
train track is a straight one.
Slide 6: Distance travelled is always
positive. It is cumulative and so can never
‘drop back down’ (have a negative gradient).
Make it ‘girl friendly’:
Ask students to
discuss it with a friend
before giving an answer
Teacher notes: Displacement and distance
Slides 10 & 11: It’s really worth ensuring
that students understand the difference
between the 3 terms and appreciate how
the graphs correspond to each other.
Ensure that students realise that
distance travelled is cumulative.
Distance and distance travelled are
always positive quantities; displacement
can be positive or negative
Slides 12 - 15: An A4 sheet of 9
matching cards (Distance and
Displacement graphs) is available as an
alternative to using the slides.
Make it ‘girl friendly’:
Print out the matching
cards and ask students to
work in pairs.
Teacher notes: Distance and Displacement
Answers:
Slides 12 & 13: 2, 5 and 6
Slides 14 & 15: 1 & 3 have the same
distance time graph (below); the second
one is for displacement graph 4
Matching cards answers:
A with E, H and J
B with D and F
C has no match
• can you draw one? (Yes)
• Is there more than once
answer for this? (Yes,
infinite possibilities).
G also has no match:
• can you draw one?
(Yes).
• Is there more than one
answer for this? (No: this
graph only).
Teacher notes: Speed and velocity
Slide 21 : Speed is always positive, velocity can be positive or negative.
Slide 24: The gradient of the last part is negative, but since speed is a
scalar, it can’t be negative. This is a big issue with ‘distance from’ – time
graphs, which are often used.
Slide 25 :It can make a difference, sometimes the graphs will look the same
and the values will be the same, often they’re not.
• When are they the same?
Slide 26: An example of when the values are
different.
Slide 28: Either show the slide, use the cards or
print out as a worksheet. Cards would be easier
for students as they will be able to annotate
them. Answers on final slide.
Make it ‘girl
friendly’:
Cut out the cards and
work with a partner.
Average velocity is 0m/s
Average velocity is1.33m/s
Average velocity is -1m/s
Average velocity is -1m/s
Distance travelled is 12m
Distance travelled is 14m
Distance travelled is 14m
Distance travelled is 16m
Average speed is 2m/s
Average speed is 2.33m/s
Average speed is 2.33m/s
Average speed is 2.66m/s